A Butterfly-Accelerated Volume Integral Equation Solver for Broad Permittivity and Large-Scale Electromagnetic Analysis

TL;DR

This paper introduces a butterfly-accelerated volume integral equation solver that significantly reduces computational complexity and memory usage for large-scale electromagnetic scattering problems involving heterogeneous objects with broad permittivity.

Contribution

It develops a hierarchical off-diagonal butterfly scheme for efficient matrix construction and inversion, enabling faster and more memory-efficient EM analysis of large, complex structures.

Findings

Achieves $O(N ext{log}^2N)$ matrix construction cost.

Achieves $O(N^{1.5} ext{log}N)$ inversion cost.

Demonstrates effectiveness on large-scale real-world structures.

Abstract

A butterfly-accelerated volume integral equation (VIE) solver is proposed for fast and accurate electromagnetic (EM) analysis of scattering from heterogeneous objects. The proposed solver leverages the hierarchical off-diagonal butterfly (HOD-BF) scheme to construct the system matrix and obtain its approximate inverse, used as a preconditioner. Complexity analysis and numerical experiments validate the $O(N\log^2N)$ construction cost of the HOD-BF-compressed system matrix and $O(N^{1.5}\log N)$ inversion cost for the preconditioner, where $N$ is the number of unknowns in the high-frequency EM scattering problem. For many practical scenarios, the proposed VIE solver requires less memory and computational time to construct the system matrix and obtain its approximate inverse compared to a $\mathcal{H}$ matrix-accelerated VIE solver. The accuracy and efficiency of the proposed solver have been demonstrated via its application to the EM analysis of large-scale canonical and real-world structures comprising of broad permittivity values and involving millions of unknowns.